∫ e−4x(−5120+37888x−102400x2+114688x3−32768x4−16384x5−4e4xx16+exx2(−4096x2+22272x3−38400x4+17408x5+6144x6)+e2xx4(−1152x4+4032x5−3072x6−768x7)+e3xx6(−128x6+208x7+32x8))_______________________________________________________________________________________________________________________________________________

Optimal. Leaf size=25 \[ \frac {\left (\frac {2 e^{-x} \left (-4+\frac {2}{x}\right )}{x^2}+x\right )^4}{x^8} \]

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Rubi [B] time = 16.02, antiderivative size = 144, normalized size of antiderivative = 5.76, number of steps used = 268, number of rules used = 6, integrand size = 132, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.045, Rules used = {6688, 12, 6742, 2199, 2177, 2178} \begin {gather*} \frac {256 e^{-4 x}}{x^{20}}-\frac {2048 e^{-4 x}}{x^{19}}+\frac {6144 e^{-4 x}}{x^{18}}-\frac {8192 e^{-4 x}}{x^{17}}+\frac {4096 e^{-4 x}}{x^{16}}+\frac {256 e^{-3 x}}{x^{16}}-\frac {1536 e^{-3 x}}{x^{15}}+\frac {3072 e^{-3 x}}{x^{14}}-\frac {2048 e^{-3 x}}{x^{13}}+\frac {96 e^{-2 x}}{x^{12}}-\frac {384 e^{-2 x}}{x^{11}}+\frac {384 e^{-2 x}}{x^{10}}+\frac {16 e^{-x}}{x^8}-\frac {32 e^{-x}}{x^7}+\frac {1}{x^4} \end {gather*}

Int[(-5120 + 37888*x - 102400*x^2 + 114688*x^3 - 32768*x^4 - 16384*x^5 - 4*E^(4*x)*x^16 + E^x*x^2*(-4096*x^2 +

22272*x^3 - 38400*x^4 + 17408*x^5 + 6144*x^6) + E^(2*x)*x^4*(-1152*x^4 + 4032*x^5 - 3072*x^6 - 768*x^7) + E^(

256/(E^(4*x)*x^20) - 2048/(E^(4*x)*x^19) + 6144/(E^(4*x)*x^18) - 8192/(E^(4*x)*x^17) + 4096/(E^(4*x)*x^16) + 2

56/(E^(3*x)*x^16) - 1536/(E^(3*x)*x^15) + 3072/(E^(3*x)*x^14) - 2048/(E^(3*x)*x^13) + 96/(E^(2*x)*x^12) - 384/

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m

+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), Int[(c + d*x)^(m + 1)*(b*F^(g*

(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] && !$UseGamma ===

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral

Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] && !$UseGamma === True

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePo

werOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x]

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 e^{-4 x} \left (-20+28 x+8 x^2-e^x x^4\right ) \left (4-8 x+e^x x^4\right )^3}{x^{21}} \, dx\\ &=4 \int \frac {e^{-4 x} \left (-20+28 x+8 x^2-e^x x^4\right ) \left (4-8 x+e^x x^4\right )^3}{x^{21}} \, dx\\ &=4 \int \left (-\frac {1}{x^5}-\frac {256 e^{-4 x} (-1+2 x)^3 \left (-5+7 x+2 x^2\right )}{x^{21}}+\frac {4 e^{-x} \left (-8+13 x+2 x^2\right )}{x^9}+\frac {64 e^{-3 x} (-1+2 x)^2 \left (-16+23 x+6 x^2\right )}{x^{17}}-\frac {48 e^{-2 x} \left (6-21 x+16 x^2+4 x^3\right )}{x^{13}}\right ) \, dx\\ &=\frac {1}{x^4}+16 \int \frac {e^{-x} \left (-8+13 x+2 x^2\right )}{x^9} \, dx-192 \int \frac {e^{-2 x} \left (6-21 x+16 x^2+4 x^3\right )}{x^{13}} \, dx+256 \int \frac {e^{-3 x} (-1+2 x)^2 \left (-16+23 x+6 x^2\right )}{x^{17}} \, dx-1024 \int \frac {e^{-4 x} (-1+2 x)^3 \left (-5+7 x+2 x^2\right )}{x^{21}} \, dx\\ &=\frac {1}{x^4}+16 \int \left (-\frac {8 e^{-x}}{x^9}+\frac {13 e^{-x}}{x^8}+\frac {2 e^{-x}}{x^7}\right ) \, dx-192 \int \left (\frac {6 e^{-2 x}}{x^{13}}-\frac {21 e^{-2 x}}{x^{12}}+\frac {16 e^{-2 x}}{x^{11}}+\frac {4 e^{-2 x}}{x^{10}}\right ) \, dx+256 \int \left (-\frac {16 e^{-3 x}}{x^{17}}+\frac {87 e^{-3 x}}{x^{16}}-\frac {150 e^{-3 x}}{x^{15}}+\frac {68 e^{-3 x}}{x^{14}}+\frac {24 e^{-3 x}}{x^{13}}\right ) \, dx-1024 \int \left (\frac {5 e^{-4 x}}{x^{21}}-\frac {37 e^{-4 x}}{x^{20}}+\frac {100 e^{-4 x}}{x^{19}}-\frac {112 e^{-4 x}}{x^{18}}+\frac {32 e^{-4 x}}{x^{17}}+\frac {16 e^{-4 x}}{x^{16}}\right ) \, dx\\ &=\frac {1}{x^4}+32 \int \frac {e^{-x}}{x^7} \, dx-128 \int \frac {e^{-x}}{x^9} \, dx+208 \int \frac {e^{-x}}{x^8} \, dx-768 \int \frac {e^{-2 x}}{x^{10}} \, dx-1152 \int \frac {e^{-2 x}}{x^{13}} \, dx-3072 \int \frac {e^{-2 x}}{x^{11}} \, dx+4032 \int \frac {e^{-2 x}}{x^{12}} \, dx-4096 \int \frac {e^{-3 x}}{x^{17}} \, dx-5120 \int \frac {e^{-4 x}}{x^{21}} \, dx+6144 \int \frac {e^{-3 x}}{x^{13}} \, dx-16384 \int \frac {e^{-4 x}}{x^{16}} \, dx+17408 \int \frac {e^{-3 x}}{x^{14}} \, dx+22272 \int \frac {e^{-3 x}}{x^{16}} \, dx-32768 \int \frac {e^{-4 x}}{x^{17}} \, dx+37888 \int \frac {e^{-4 x}}{x^{20}} \, dx-38400 \int \frac {e^{-3 x}}{x^{15}} \, dx-102400 \int \frac {e^{-4 x}}{x^{19}} \, dx+114688 \int \frac {e^{-4 x}}{x^{18}} \, dx\\ &=\frac {256 e^{-4 x}}{x^{20}}-\frac {37888 e^{-4 x}}{19 x^{19}}+\frac {51200 e^{-4 x}}{9 x^{18}}-\frac {114688 e^{-4 x}}{17 x^{17}}+\frac {2048 e^{-4 x}}{x^{16}}+\frac {256 e^{-3 x}}{x^{16}}+\frac {16384 e^{-4 x}}{15 x^{15}}-\frac {7424 e^{-3 x}}{5 x^{15}}+\frac {19200 e^{-3 x}}{7 x^{14}}-\frac {17408 e^{-3 x}}{13 x^{13}}-\frac {512 e^{-3 x}}{x^{12}}+\frac {96 e^{-2 x}}{x^{12}}-\frac {4032 e^{-2 x}}{11 x^{11}}+\frac {1536 e^{-2 x}}{5 x^{10}}+\frac {256 e^{-2 x}}{3 x^9}+\frac {16 e^{-x}}{x^8}-\frac {208 e^{-x}}{7 x^7}-\frac {16 e^{-x}}{3 x^6}+\frac {1}{x^4}-\frac {16}{3} \int \frac {e^{-x}}{x^6} \, dx+16 \int \frac {e^{-x}}{x^8} \, dx-\frac {208}{7} \int \frac {e^{-x}}{x^7} \, dx+\frac {512}{3} \int \frac {e^{-2 x}}{x^9} \, dx+192 \int \frac {e^{-2 x}}{x^{12}} \, dx+\frac {3072}{5} \int \frac {e^{-2 x}}{x^{10}} \, dx-\frac {8064}{11} \int \frac {e^{-2 x}}{x^{11}} \, dx+768 \int \frac {e^{-3 x}}{x^{16}} \, dx+1024 \int \frac {e^{-4 x}}{x^{20}} \, dx-1536 \int \frac {e^{-3 x}}{x^{12}} \, dx-\frac {52224}{13} \int \frac {e^{-3 x}}{x^{13}} \, dx+\frac {65536}{15} \int \frac {e^{-4 x}}{x^{15}} \, dx-\frac {22272}{5} \int \frac {e^{-3 x}}{x^{15}} \, dx-\frac {151552}{19} \int \frac {e^{-4 x}}{x^{19}} \, dx+8192 \int \frac {e^{-4 x}}{x^{16}} \, dx+\frac {57600}{7} \int \frac {e^{-3 x}}{x^{14}} \, dx+\frac {204800}{9} \int \frac {e^{-4 x}}{x^{18}} \, dx-\frac {458752}{17} \int \frac {e^{-4 x}}{x^{17}} \, dx\\ &=\frac {256 e^{-4 x}}{x^{20}}-\frac {2048 e^{-4 x}}{x^{19}}+\frac {1048576 e^{-4 x}}{171 x^{18}}-\frac {1236992 e^{-4 x}}{153 x^{17}}+\frac {63488 e^{-4 x}}{17 x^{16}}+\frac {256 e^{-3 x}}{x^{16}}+\frac {8192 e^{-4 x}}{15 x^{15}}-\frac {1536 e^{-3 x}}{x^{15}}-\frac {32768 e^{-4 x}}{105 x^{14}}+\frac {107136 e^{-3 x}}{35 x^{14}}-\frac {179456 e^{-3 x}}{91 x^{13}}-\frac {2304 e^{-3 x}}{13 x^{12}}+\frac {96 e^{-2 x}}{x^{12}}+\frac {1536 e^{-3 x}}{11 x^{11}}-\frac {384 e^{-2 x}}{x^{11}}+\frac {20928 e^{-2 x}}{55 x^{10}}+\frac {256 e^{-2 x}}{15 x^9}-\frac {64 e^{-2 x}}{3 x^8}+\frac {16 e^{-x}}{x^8}-\frac {32 e^{-x}}{x^7}-\frac {8 e^{-x}}{21 x^6}+\frac {16 e^{-x}}{15 x^5}+\frac {1}{x^4}+\frac {16}{15} \int \frac {e^{-x}}{x^5} \, dx-\frac {16}{7} \int \frac {e^{-x}}{x^7} \, dx+\frac {104}{21} \int \frac {e^{-x}}{x^6} \, dx-\frac {384}{11} \int \frac {e^{-2 x}}{x^{11}} \, dx-\frac {128}{3} \int \frac {e^{-2 x}}{x^8} \, dx-\frac {2048}{15} \int \frac {e^{-2 x}}{x^9} \, dx+\frac {8064}{55} \int \frac {e^{-2 x}}{x^{10}} \, dx-\frac {768}{5} \int \frac {e^{-3 x}}{x^{15}} \, dx-\frac {4096}{19} \int \frac {e^{-4 x}}{x^{19}} \, dx+\frac {4608}{11} \int \frac {e^{-3 x}}{x^{11}} \, dx+\frac {33408}{35} \int \frac {e^{-3 x}}{x^{14}} \, dx+\frac {13056}{13} \int \frac {e^{-3 x}}{x^{12}} \, dx-\frac {131072}{105} \int \frac {e^{-4 x}}{x^{14}} \, dx+\frac {303104}{171} \int \frac {e^{-4 x}}{x^{18}} \, dx-\frac {172800}{91} \int \frac {e^{-3 x}}{x^{13}} \, dx-\frac {32768}{15} \int \frac {e^{-4 x}}{x^{15}} \, dx-\frac {819200}{153} \int \frac {e^{-4 x}}{x^{17}} \, dx+\frac {114688}{17} \int \frac {e^{-4 x}}{x^{16}} \, dx\\ &=\frac {256 e^{-4 x}}{x^{20}}-\frac {2048 e^{-4 x}}{x^{19}}+\frac {6144 e^{-4 x}}{x^{18}}-\frac {23805952 e^{-4 x}}{2907 x^{17}}+\frac {622592 e^{-4 x}}{153 x^{16}}+\frac {256 e^{-3 x}}{x^{16}}+\frac {8192 e^{-4 x}}{85 x^{15}}-\frac {1536 e^{-3 x}}{x^{15}}-\frac {16384 e^{-4 x}}{105 x^{14}}+\frac {3072 e^{-3 x}}{x^{14}}+\frac {131072 e^{-4 x}}{1365 x^{13}}-\frac {930688 e^{-3 x}}{455 x^{13}}-\frac {1728 e^{-3 x}}{91 x^{12}}+\frac {96 e^{-2 x}}{x^{12}}+\frac {6912 e^{-3 x}}{143 x^{11}}-\frac {384 e^{-2 x}}{x^{11}}-\frac {2304 e^{-3 x}}{55 x^{10}}+\frac {384 e^{-2 x}}{x^{10}}+\frac {128 e^{-2 x}}{165 x^9}-\frac {64 e^{-2 x}}{15 x^8}+\frac {16 e^{-x}}{x^8}+\frac {128 e^{-2 x}}{21 x^7}-\frac {32 e^{-x}}{x^7}+\frac {8 e^{-x}}{105 x^5}+\frac {1}{x^4}-\frac {4 e^{-x}}{15 x^4}-\frac {4}{15} \int \frac {e^{-x}}{x^4} \, dx+\frac {8}{21} \int \frac {e^{-x}}{x^6} \, dx-\frac {104}{105} \int \frac {e^{-x}}{x^5} \, dx+\frac {384}{55} \int \frac {e^{-2 x}}{x^{10}} \, dx+\frac {256}{21} \int \frac {e^{-2 x}}{x^7} \, dx-\frac {1792}{55} \int \frac {e^{-2 x}}{x^9} \, dx+\frac {1152}{35} \int \frac {e^{-3 x}}{x^{14}} \, dx+\frac {512}{15} \int \frac {e^{-2 x}}{x^8} \, dx+\frac {8192}{171} \int \frac {e^{-4 x}}{x^{18}} \, dx-\frac {6912}{55} \int \frac {e^{-3 x}}{x^{10}} \, dx-\frac {100224}{455} \int \frac {e^{-3 x}}{x^{13}} \, dx-\frac {39168}{143} \int \frac {e^{-3 x}}{x^{11}} \, dx+\frac {524288 \int \frac {e^{-4 x}}{x^{13}} \, dx}{1365}-\frac {1212416 \int \frac {e^{-4 x}}{x^{17}} \, dx}{2907}+\frac {43200}{91} \int \frac {e^{-3 x}}{x^{12}} \, dx+\frac {65536}{105} \int \frac {e^{-4 x}}{x^{14}} \, dx+\frac {204800}{153} \int \frac {e^{-4 x}}{x^{16}} \, dx-\frac {458752}{255} \int \frac {e^{-4 x}}{x^{15}} \, dx\\ &=\frac {256 e^{-4 x}}{x^{20}}-\frac {2048 e^{-4 x}}{x^{19}}+\frac {6144 e^{-4 x}}{x^{18}}-\frac {8192 e^{-4 x}}{x^{17}}+\frac {11905024 e^{-4 x}}{2907 x^{16}}+\frac {256 e^{-3 x}}{x^{16}}+\frac {16384 e^{-4 x}}{2295 x^{15}}-\frac {1536 e^{-3 x}}{x^{15}}-\frac {16384 e^{-4 x}}{595 x^{14}}+\frac {3072 e^{-3 x}}{x^{14}}+\frac {65536 e^{-4 x}}{1365 x^{13}}-\frac {2048 e^{-3 x}}{x^{13}}-\frac {131072 e^{-4 x}}{4095 x^{12}}-\frac {288 e^{-3 x}}{455 x^{12}}+\frac {96 e^{-2 x}}{x^{12}}+\frac {5184 e^{-3 x}}{1001 x^{11}}-\frac {384 e^{-2 x}}{x^{11}}-\frac {10368 e^{-3 x}}{715 x^{10}}+\frac {384 e^{-2 x}}{x^{10}}+\frac {768 e^{-3 x}}{55 x^9}-\frac {32 e^{-2 x}}{165 x^8}+\frac {16 e^{-x}}{x^8}+\frac {128 e^{-2 x}}{105 x^7}-\frac {32 e^{-x}}{x^7}-\frac {128 e^{-2 x}}{63 x^6}+\frac {1}{x^4}-\frac {2 e^{-x}}{105 x^4}+\frac {4 e^{-x}}{45 x^3}-\frac {8}{105} \int \frac {e^{-x}}{x^5} \, dx+\frac {4}{45} \int \frac {e^{-x}}{x^3} \, dx+\frac {26}{105} \int \frac {e^{-x}}{x^4} \, dx-\frac {256}{165} \int \frac {e^{-2 x}}{x^9} \, dx-\frac {256}{63} \int \frac {e^{-2 x}}{x^6} \, dx-\frac {3456}{455} \int \frac {e^{-3 x}}{x^{13}} \, dx+\frac {448}{55} \int \frac {e^{-2 x}}{x^8} \, dx-\frac {1024}{105} \int \frac {e^{-2 x}}{x^7} \, dx-\frac {32768 \int \frac {e^{-4 x}}{x^{17}} \, dx}{2907}+\frac {2304}{55} \int \frac {e^{-3 x}}{x^9} \, dx+\frac {25056}{455} \int \frac {e^{-3 x}}{x^{12}} \, dx+\frac {58752}{715} \int \frac {e^{-3 x}}{x^{10}} \, dx+\frac {303104 \int \frac {e^{-4 x}}{x^{16}} \, dx}{2907}-\frac {524288 \int \frac {e^{-4 x}}{x^{12}} \, dx}{4095}-\frac {129600 \int \frac {e^{-3 x}}{x^{11}} \, dx}{1001}-\frac {262144 \int \frac {e^{-4 x}}{x^{13}} \, dx}{1365}-\frac {163840}{459} \int \frac {e^{-4 x}}{x^{15}} \, dx+\frac {131072}{255} \int \frac {e^{-4 x}}{x^{14}} \, dx\\ &=\frac {256 e^{-4 x}}{x^{20}}-\frac {2048 e^{-4 x}}{x^{19}}+\frac {6144 e^{-4 x}}{x^{18}}-\frac {8192 e^{-4 x}}{x^{17}}+\frac {4096 e^{-4 x}}{x^{16}}+\frac {256 e^{-3 x}}{x^{16}}+\frac {8192 e^{-4 x}}{43605 x^{15}}-\frac {1536 e^{-3 x}}{x^{15}}-\frac {32768 e^{-4 x}}{16065 x^{14}}+\frac {3072 e^{-3 x}}{x^{14}}+\frac {65536 e^{-4 x}}{7735 x^{13}}-\frac {2048 e^{-3 x}}{x^{13}}-\frac {65536 e^{-4 x}}{4095 x^{12}}+\frac {96 e^{-2 x}}{x^{12}}+\frac {524288 e^{-4 x}}{45045 x^{11}}+\frac {864 e^{-3 x}}{5005 x^{11}}-\frac {384 e^{-2 x}}{x^{11}}-\frac {7776 e^{-3 x}}{5005 x^{10}}+\frac {384 e^{-2 x}}{x^{10}}+\frac {3456 e^{-3 x}}{715 x^9}-\frac {288 e^{-3 x}}{55 x^8}+\frac {16 e^{-x}}{x^8}+\frac {64 e^{-2 x}}{1155 x^7}-\frac {32 e^{-x}}{x^7}-\frac {128 e^{-2 x}}{315 x^6}+\frac {256 e^{-2 x}}{315 x^5}+\frac {1}{x^4}+\frac {2 e^{-x}}{315 x^3}-\frac {2 e^{-x}}{45 x^2}+\frac {2}{105} \int \frac {e^{-x}}{x^4} \, dx-\frac {2}{45} \int \frac {e^{-x}}{x^2} \, dx-\frac {26}{315} \int \frac {e^{-x}}{x^3} \, dx+\frac {64}{165} \int \frac {e^{-2 x}}{x^8} \, dx+\frac {512}{315} \int \frac {e^{-2 x}}{x^5} \, dx+\frac {864}{455} \int \frac {e^{-3 x}}{x^{12}} \, dx-\frac {128}{55} \int \frac {e^{-2 x}}{x^7} \, dx+\frac {8192 \int \frac {e^{-4 x}}{x^{16}} \, dx}{2907}+\frac {1024}{315} \int \frac {e^{-2 x}}{x^6} \, dx-\frac {75168 \int \frac {e^{-3 x}}{x^{11}} \, dx}{5005}-\frac {864}{55} \int \frac {e^{-3 x}}{x^8} \, dx-\frac {19584}{715} \int \frac {e^{-3 x}}{x^9} \, dx-\frac {1212416 \int \frac {e^{-4 x}}{x^{15}} \, dx}{43605}+\frac {38880 \int \frac {e^{-3 x}}{x^{10}} \, dx}{1001}+\frac {2097152 \int \frac {e^{-4 x}}{x^{11}} \, dx}{45045}+\frac {262144 \int \frac {e^{-4 x}}{x^{12}} \, dx}{4095}+\frac {327680 \int \frac {e^{-4 x}}{x^{14}} \, dx}{3213}-\frac {524288 \int \frac {e^{-4 x}}{x^{13}} \, dx}{3315}\\ &=\frac {256 e^{-4 x}}{x^{20}}-\frac {2048 e^{-4 x}}{x^{19}}+\frac {6144 e^{-4 x}}{x^{18}}-\frac {8192 e^{-4 x}}{x^{17}}+\frac {4096 e^{-4 x}}{x^{16}}+\frac {256 e^{-3 x}}{x^{16}}-\frac {1536 e^{-3 x}}{x^{15}}-\frac {16384 e^{-4 x}}{305235 x^{14}}+\frac {3072 e^{-3 x}}{x^{14}}+\frac {131072 e^{-4 x}}{208845 x^{13}}-\frac {2048 e^{-3 x}}{x^{13}}-\frac {65536 e^{-4 x}}{23205 x^{12}}+\frac {96 e^{-2 x}}{x^{12}}+\frac {262144 e^{-4 x}}{45045 x^{11}}-\frac {384 e^{-2 x}}{x^{11}}-\frac {1048576 e^{-4 x}}{225225 x^{10}}-\frac {1296 e^{-3 x}}{25025 x^{10}}+\frac {384 e^{-2 x}}{x^{10}}+\frac {2592 e^{-3 x}}{5005 x^9}-\frac {1296 e^{-3 x}}{715 x^8}+\frac {16 e^{-x}}{x^8}+\frac {864 e^{-3 x}}{385 x^7}-\frac {32 e^{-x}}{x^7}-\frac {64 e^{-2 x}}{3465 x^6}+\frac {256 e^{-2 x}}{1575 x^5}+\frac {1}{x^4}-\frac {128 e^{-2 x}}{315 x^4}-\frac {e^{-x}}{315 x^2}+\frac {2 e^{-x}}{45 x}-\frac {2}{315} \int \frac {e^{-x}}{x^3} \, dx+\frac {13}{315} \int \frac {e^{-x}}{x^2} \, dx+\frac {2}{45} \int \frac {e^{-x}}{x} \, dx-\frac {128 \int \frac {e^{-2 x}}{x^7} \, dx}{1155}-\frac {2592 \int \frac {e^{-3 x}}{x^{11}} \, dx}{5005}-\frac {32768 \int \frac {e^{-4 x}}{x^{15}} \, dx}{43605}+\frac {128}{165} \int \frac {e^{-2 x}}{x^6} \, dx-\frac {256}{315} \int \frac {e^{-2 x}}{x^4} \, dx-\frac {2048 \int \frac {e^{-2 x}}{x^5} \, dx}{1575}+\frac {112752 \int \frac {e^{-3 x}}{x^{10}} \, dx}{25025}+\frac {2592}{385} \int \frac {e^{-3 x}}{x^7} \, dx+\frac {2424832 \int \frac {e^{-4 x}}{x^{14}} \, dx}{305235}+\frac {7344}{715} \int \frac {e^{-3 x}}{x^8} \, dx-\frac {12960 \int \frac {e^{-3 x}}{x^9} \, dx}{1001}-\frac {4194304 \int \frac {e^{-4 x}}{x^{10}} \, dx}{225225}-\frac {1048576 \int \frac {e^{-4 x}}{x^{11}} \, dx}{45045}-\frac {1310720 \int \frac {e^{-4 x}}{x^{13}} \, dx}{41769}+\frac {524288 \int \frac {e^{-4 x}}{x^{12}} \, dx}{9945}\\ &=\frac {256 e^{-4 x}}{x^{20}}-\frac {2048 e^{-4 x}}{x^{19}}+\frac {6144 e^{-4 x}}{x^{18}}-\frac {8192 e^{-4 x}}{x^{17}}+\frac {4096 e^{-4 x}}{x^{16}}+\frac {256 e^{-3 x}}{x^{16}}-\frac {1536 e^{-3 x}}{x^{15}}+\frac {3072 e^{-3 x}}{x^{14}}+\frac {65536 e^{-4 x}}{3968055 x^{13}}-\frac {2048 e^{-3 x}}{x^{13}}-\frac {131072 e^{-4 x}}{626535 x^{12}}+\frac {96 e^{-2 x}}{x^{12}}+\frac {262144 e^{-4 x}}{255255 x^{11}}-\frac {384 e^{-2 x}}{x^{11}}-\frac {524288 e^{-4 x}}{225225 x^{10}}+\frac {384 e^{-2 x}}{x^{10}}+\frac {4194304 e^{-4 x}}{2027025 x^9}+\frac {432 e^{-3 x}}{25025 x^9}-\frac {972 e^{-3 x}}{5005 x^8}+\frac {16 e^{-x}}{x^8}+\frac {3888 e^{-3 x}}{5005 x^7}-\frac {32 e^{-x}}{x^7}-\frac {432 e^{-3 x}}{385 x^6}+\frac {128 e^{-2 x}}{17325 x^5}+\frac {1}{x^4}-\frac {128 e^{-2 x}}{1575 x^4}+\frac {256 e^{-2 x}}{945 x^3}+\frac {e^{-x}}{315 x}+\frac {2 \text {Ei}(-x)}{45}+\frac {1}{315} \int \frac {e^{-x}}{x^2} \, dx+\frac {128 \int \frac {e^{-2 x}}{x^6} \, dx}{3465}-\frac {13}{315} \int \frac {e^{-x}}{x} \, dx+\frac {3888 \int \frac {e^{-3 x}}{x^{10}} \, dx}{25025}+\frac {65536 \int \frac {e^{-4 x}}{x^{14}} \, dx}{305235}-\frac {256}{825} \int \frac {e^{-2 x}}{x^5} \, dx+\frac {512}{945} \int \frac {e^{-2 x}}{x^3} \, dx+\frac {1024 \int \frac {e^{-2 x}}{x^4} \, dx}{1575}-\frac {37584 \int \frac {e^{-3 x}}{x^9} \, dx}{25025}-\frac {9699328 \int \frac {e^{-4 x}}{x^{13}} \, dx}{3968055}-\frac {1296}{385} \int \frac {e^{-3 x}}{x^6} \, dx-\frac {22032 \int \frac {e^{-3 x}}{x^7} \, dx}{5005}+\frac {4860 \int \frac {e^{-3 x}}{x^8} \, dx}{1001}+\frac {16777216 \int \frac {e^{-4 x}}{x^9} \, dx}{2027025}+\frac {2097152 \int \frac {e^{-4 x}}{x^{10}} \, dx}{225225}+\frac {1310720 \int \frac {e^{-4 x}}{x^{12}} \, dx}{125307}-\frac {2097152 \int \frac {e^{-4 x}}{x^{11}} \, dx}{109395}\\ &=\frac {256 e^{-4 x}}{x^{20}}-\frac {2048 e^{-4 x}}{x^{19}}+\frac {6144 e^{-4 x}}{x^{18}}-\frac {8192 e^{-4 x}}{x^{17}}+\frac {4096 e^{-4 x}}{x^{16}}+\frac {256 e^{-3 x}}{x^{16}}-\frac {1536 e^{-3 x}}{x^{15}}+\frac {3072 e^{-3 x}}{x^{14}}-\frac {2048 e^{-3 x}}{x^{13}}-\frac {65536 e^{-4 x}}{11904165 x^{12}}+\frac {96 e^{-2 x}}{x^{12}}+\frac {524288 e^{-4 x}}{6891885 x^{11}}-\frac {384 e^{-2 x}}{x^{11}}-\frac {524288 e^{-4 x}}{1276275 x^{10}}+\frac {384 e^{-2 x}}{x^{10}}+\frac {2097152 e^{-4 x}}{2027025 x^9}-\frac {2097152 e^{-4 x}}{2027025 x^8}-\frac {162 e^{-3 x}}{25025 x^8}+\frac {16 e^{-x}}{x^8}+\frac {2916 e^{-3 x}}{35035 x^7}-\frac {32 e^{-x}}{x^7}-\frac {1944 e^{-3 x}}{5005 x^6}+\frac {1296 e^{-3 x}}{1925 x^5}+\frac {1}{x^4}-\frac {64 e^{-2 x}}{17325 x^4}+\frac {256 e^{-2 x}}{4725 x^3}-\frac {256 e^{-2 x}}{945 x^2}+\frac {\text {Ei}(-x)}{315}-\frac {1}{315} \int \frac {e^{-x}}{x} \, dx-\frac {256 \int \frac {e^{-2 x}}{x^5} \, dx}{17325}-\frac {1296 \int \frac {e^{-3 x}}{x^9} \, dx}{25025}-\frac {262144 \int \frac {e^{-4 x}}{x^{13}} \, dx}{3968055}+\frac {128}{825} \int \frac {e^{-2 x}}{x^4} \, dx-\frac {2048 \int \frac {e^{-2 x}}{x^3} \, dx}{4725}-\frac {512}{945} \int \frac {e^{-2 x}}{x^2} \, dx+\frac {14094 \int \frac {e^{-3 x}}{x^8} \, dx}{25025}+\frac {9699328 \int \frac {e^{-4 x}}{x^{12}} \, dx}{11904165}+\frac {3888 \int \frac {e^{-3 x}}{x^5} \, dx}{1925}-\frac {14580 \int \frac {e^{-3 x}}{x^7} \, dx}{7007}+\frac {11016 \int \frac {e^{-3 x}}{x^6} \, dx}{5005}-\frac {5242880 \int \frac {e^{-4 x}}{x^{11}} \, dx}{1378377}-\frac {8388608 \int \frac {e^{-4 x}}{x^9} \, dx}{2027025}-\frac {8388608 \int \frac {e^{-4 x}}{x^8} \, dx}{2027025}+\frac {4194304 \int \frac {e^{-4 x}}{x^{10}} \, dx}{546975}\\ &=\frac {256 e^{-4 x}}{x^{20}}-\frac {2048 e^{-4 x}}{x^{19}}+\frac {6144 e^{-4 x}}{x^{18}}-\frac {8192 e^{-4 x}}{x^{17}}+\frac {4096 e^{-4 x}}{x^{16}}+\frac {256 e^{-3 x}}{x^{16}}-\frac {1536 e^{-3 x}}{x^{15}}+\frac {3072 e^{-3 x}}{x^{14}}-\frac {2048 e^{-3 x}}{x^{13}}+\frac {96 e^{-2 x}}{x^{12}}+\frac {262144 e^{-4 x}}{130945815 x^{11}}-\frac {384 e^{-2 x}}{x^{11}}-\frac {1048576 e^{-4 x}}{34459425 x^{10}}+\frac {384 e^{-2 x}}{x^{10}}+\frac {2097152 e^{-4 x}}{11486475 x^9}-\frac {1048576 e^{-4 x}}{2027025 x^8}+\frac {16 e^{-x}}{x^8}+\frac {8388608 e^{-4 x}}{14189175 x^7}+\frac {486 e^{-3 x}}{175175 x^7}-\frac {32 e^{-x}}{x^7}-\frac {1458 e^{-3 x}}{35035 x^6}+\frac {5832 e^{-3 x}}{25025 x^5}+\frac {1}{x^4}-\frac {972 e^{-3 x}}{1925 x^4}+\frac {128 e^{-2 x}}{51975 x^3}-\frac {256 e^{-2 x}}{4725 x^2}+\frac {512 e^{-2 x}}{945 x}+\frac {128 \int \frac {e^{-2 x}}{x^4} \, dx}{17325}+\frac {486 \int \frac {e^{-3 x}}{x^8} \, dx}{25025}+\frac {262144 \int \frac {e^{-4 x}}{x^{12}} \, dx}{11904165}-\frac {256 \int \frac {e^{-2 x}}{x^3} \, dx}{2475}-\frac {42282 \int \frac {e^{-3 x}}{x^7} \, dx}{175175}-\frac {38797312 \int \frac {e^{-4 x}}{x^{11}} \, dx}{130945815}+\frac {2048 \int \frac {e^{-2 x}}{x^2} \, dx}{4725}+\frac {7290 \int \frac {e^{-3 x}}{x^6} \, dx}{7007}+\frac {1024}{945} \int \frac {e^{-2 x}}{x} \, dx-\frac {33048 \int \frac {e^{-3 x}}{x^5} \, dx}{25025}-\frac {2916 \int \frac {e^{-3 x}}{x^4} \, dx}{1925}+\frac {2097152 \int \frac {e^{-4 x}}{x^{10}} \, dx}{1378377}+\frac {4194304 \int \frac {e^{-4 x}}{x^8} \, dx}{2027025}+\frac {33554432 \int \frac {e^{-4 x}}{x^7} \, dx}{14189175}-\frac {16777216 \int \frac {e^{-4 x}}{x^9} \, dx}{4922775}\\ &=\frac {256 e^{-4 x}}{x^{20}}-\frac {2048 e^{-4 x}}{x^{19}}+\frac {6144 e^{-4 x}}{x^{18}}-\frac {8192 e^{-4 x}}{x^{17}}+\frac {4096 e^{-4 x}}{x^{16}}+\frac {256 e^{-3 x}}{x^{16}}-\frac {1536 e^{-3 x}}{x^{15}}+\frac {3072 e^{-3 x}}{x^{14}}-\frac {2048 e^{-3 x}}{x^{13}}+\frac {96 e^{-2 x}}{x^{12}}-\frac {384 e^{-2 x}}{x^{11}}-\frac {524288 e^{-4 x}}{654729075 x^{10}}+\frac {384 e^{-2 x}}{x^{10}}+\frac {4194304 e^{-4 x}}{310134825 x^9}-\frac {1048576 e^{-4 x}}{11486475 x^8}+\frac {16 e^{-x}}{x^8}+\frac {4194304 e^{-4 x}}{14189175 x^7}-\frac {32 e^{-x}}{x^7}-\frac {16777216 e^{-4 x}}{42567525 x^6}-\frac {243 e^{-3 x}}{175175 x^6}+\frac {4374 e^{-3 x}}{175175 x^5}+\frac {1}{x^4}-\frac {4374 e^{-3 x}}{25025 x^4}+\frac {972 e^{-3 x}}{1925 x^3}-\frac {128 e^{-2 x}}{51975 x^2}+\frac {512 e^{-2 x}}{4725 x}+\frac {1024 \text {Ei}(-2 x)}{945}-\frac {256 \int \frac {e^{-2 x}}{x^3} \, dx}{51975}-\frac {1048576 \int \frac {e^{-4 x}}{x^{11}} \, dx}{130945815}-\frac {1458 \int \frac {e^{-3 x}}{x^7} \, dx}{175175}+\frac {256 \int \frac {e^{-2 x}}{x^2} \, dx}{2475}+\frac {77594624 \int \frac {e^{-4 x}}{x^{10}} \, dx}{654729075}+\frac {21141 \int \frac {e^{-3 x}}{x^6} \, dx}{175175}-\frac {4374 \int \frac {e^{-3 x}}{x^5} \, dx}{7007}-\frac {8388608 \int \frac {e^{-4 x}}{x^9} \, dx}{12405393}-\frac {4096 \int \frac {e^{-2 x}}{x} \, dx}{4725}+\frac {24786 \int \frac {e^{-3 x}}{x^4} \, dx}{25025}-\frac {16777216 \int \frac {e^{-4 x}}{x^7} \, dx}{14189175}+\frac {2916 \int \frac {e^{-3 x}}{x^3} \, dx}{1925}-\frac {67108864 \int \frac {e^{-4 x}}{x^6} \, dx}{42567525}+\frac {8388608 \int \frac {e^{-4 x}}{x^8} \, dx}{4922775}\\ &=\frac {256 e^{-4 x}}{x^{20}}-\frac {2048 e^{-4 x}}{x^{19}}+\frac {6144 e^{-4 x}}{x^{18}}-\frac {8192 e^{-4 x}}{x^{17}}+\frac {4096 e^{-4 x}}{x^{16}}+\frac {256 e^{-3 x}}{x^{16}}-\frac {1536 e^{-3 x}}{x^{15}}+\frac {3072 e^{-3 x}}{x^{14}}-\frac {2048 e^{-3 x}}{x^{13}}+\frac {96 e^{-2 x}}{x^{12}}-\frac {384 e^{-2 x}}{x^{11}}+\frac {384 e^{-2 x}}{x^{10}}+\frac {2097152 e^{-4 x}}{5892561675 x^9}-\frac {2097152 e^{-4 x}}{310134825 x^8}+\frac {16 e^{-x}}{x^8}+\frac {4194304 e^{-4 x}}{80405325 x^7}-\frac {32 e^{-x}}{x^7}-\frac {8388608 e^{-4 x}}{42567525 x^6}+\frac {67108864 e^{-4 x}}{212837625 x^5}+\frac {729 e^{-3 x}}{875875 x^5}+\frac {1}{x^4}-\frac {6561 e^{-3 x}}{350350 x^4}+\frac {4374 e^{-3 x}}{25025 x^3}-\frac {1458 e^{-3 x}}{1925 x^2}+\frac {256 e^{-2 x}}{51975 x}+\frac {1024 \text {Ei}(-2 x)}{4725}+\frac {2097152 \int \frac {e^{-4 x}}{x^{10}} \, dx}{654729075}+\frac {729 \int \frac {e^{-3 x}}{x^6} \, dx}{175175}+\frac {256 \int \frac {e^{-2 x}}{x^2} \, dx}{51975}-\frac {310378496 \int \frac {e^{-4 x}}{x^9} \, dx}{5892561675}-\frac {63423 \int \frac {e^{-3 x}}{x^5} \, dx}{875875}-\frac {512 \int \frac {e^{-2 x}}{x} \, dx}{2475}+\frac {4194304 \int \frac {e^{-4 x}}{x^8} \, dx}{12405393}+\frac {6561 \int \frac {e^{-3 x}}{x^4} \, dx}{14014}+\frac {33554432 \int \frac {e^{-4 x}}{x^6} \, dx}{42567525}-\frac {33554432 \int \frac {e^{-4 x}}{x^7} \, dx}{34459425}-\frac {24786 \int \frac {e^{-3 x}}{x^3} \, dx}{25025}+\frac {268435456 \int \frac {e^{-4 x}}{x^5} \, dx}{212837625}-\frac {4374 \int \frac {e^{-3 x}}{x^2} \, dx}{1925}\\ &=\frac {256 e^{-4 x}}{x^{20}}-\frac {2048 e^{-4 x}}{x^{19}}+\frac {6144 e^{-4 x}}{x^{18}}-\frac {8192 e^{-4 x}}{x^{17}}+\frac {4096 e^{-4 x}}{x^{16}}+\frac {256 e^{-3 x}}{x^{16}}-\frac {1536 e^{-3 x}}{x^{15}}+\frac {3072 e^{-3 x}}{x^{14}}-\frac {2048 e^{-3 x}}{x^{13}}+\frac {96 e^{-2 x}}{x^{12}}-\frac {384 e^{-2 x}}{x^{11}}+\frac {384 e^{-2 x}}{x^{10}}-\frac {1048576 e^{-4 x}}{5892561675 x^8}+\frac {16 e^{-x}}{x^8}+\frac {8388608 e^{-4 x}}{2170943775 x^7}-\frac {32 e^{-x}}{x^7}-\frac {8388608 e^{-4 x}}{241215975 x^6}+\frac {33554432 e^{-4 x}}{212837625 x^5}+\frac {1}{x^4}-\frac {67108864 e^{-4 x}}{212837625 x^4}-\frac {2187 e^{-3 x}}{3503500 x^4}+\frac {6561 e^{-3 x}}{350350 x^3}-\frac {6561 e^{-3 x}}{25025 x^2}+\frac {4374 e^{-3 x}}{1925 x}+\frac {512 \text {Ei}(-2 x)}{51975}-\frac {8388608 \int \frac {e^{-4 x}}{x^9} \, dx}{5892561675}-\frac {2187 \int \frac {e^{-3 x}}{x^5} \, dx}{875875}-\frac {512 \int \frac {e^{-2 x}}{x} \, dx}{51975}+\frac {155189248 \int \frac {e^{-4 x}}{x^8} \, dx}{5892561675}+\frac {190269 \int \frac {e^{-3 x}}{x^4} \, dx}{3503500}-\frac {16777216 \int \frac {e^{-4 x}}{x^7} \, dx}{86837751}-\frac {6561 \int \frac {e^{-3 x}}{x^3} \, dx}{14014}-\frac {134217728 \int \frac {e^{-4 x}}{x^5} \, dx}{212837625}+\frac {67108864 \int \frac {e^{-4 x}}{x^6} \, dx}{103378275}-\frac {268435456 \int \frac {e^{-4 x}}{x^4} \, dx}{212837625}+\frac {37179 \int \frac {e^{-3 x}}{x^2} \, dx}{25025}+\frac {13122 \int \frac {e^{-3 x}}{x} \, dx}{1925}\\ &=\frac {256 e^{-4 x}}{x^{20}}-\frac {2048 e^{-4 x}}{x^{19}}+\frac {6144 e^{-4 x}}{x^{18}}-\frac {8192 e^{-4 x}}{x^{17}}+\frac {4096 e^{-4 x}}{x^{16}}+\frac {256 e^{-3 x}}{x^{16}}-\frac {1536 e^{-3 x}}{x^{15}}+\frac {3072 e^{-3 x}}{x^{14}}-\frac {2048 e^{-3 x}}{x^{13}}+\frac {96 e^{-2 x}}{x^{12}}-\frac {384 e^{-2 x}}{x^{11}}+\frac {384 e^{-2 x}}{x^{10}}+\frac {16 e^{-x}}{x^8}+\frac {4194304 e^{-4 x}}{41247931725 x^7}-\frac {32 e^{-x}}{x^7}-\frac {16777216 e^{-4 x}}{6512831325 x^6}+\frac {33554432 e^{-4 x}}{1206079875 x^5}+\frac {1}{x^4}-\frac {33554432 e^{-4 x}}{212837625 x^4}+\frac {268435456 e^{-4 x}}{638512875 x^3}+\frac {2187 e^{-3 x}}{3503500 x^3}-\frac {19683 e^{-3 x}}{700700 x^2}+\frac {19683 e^{-3 x}}{25025 x}+\frac {13122 \text {Ei}(-3 x)}{1925}+\frac {4194304 \int \frac {e^{-4 x}}{x^8} \, dx}{5892561675}+\frac {6561 \int \frac {e^{-3 x}}{x^4} \, dx}{3503500}-\frac {620756992 \int \frac {e^{-4 x}}{x^7} \, dx}{41247931725}-\frac {190269 \int \frac {e^{-3 x}}{x^3} \, dx}{3503500}+\frac {33554432 \int \frac {e^{-4 x}}{x^6} \, dx}{260513253}-\frac {268435456 \int \frac {e^{-4 x}}{x^5} \, dx}{516891375}+\frac {134217728 \int \frac {e^{-4 x}}{x^4} \, dx}{212837625}+\frac {19683 \int \frac {e^{-3 x}}{x^2} \, dx}{28028}+\frac {1073741824 \int \frac {e^{-4 x}}{x^3} \, dx}{638512875}-\frac {111537 \int \frac {e^{-3 x}}{x} \, dx}{25025}\\ &=\frac {256 e^{-4 x}}{x^{20}}-\frac {2048 e^{-4 x}}{x^{19}}+\frac {6144 e^{-4 x}}{x^{18}}-\frac {8192 e^{-4 x}}{x^{17}}+\frac {4096 e^{-4 x}}{x^{16}}+\frac {256 e^{-3 x}}{x^{16}}-\frac {1536 e^{-3 x}}{x^{15}}+\frac {3072 e^{-3 x}}{x^{14}}-\frac {2048 e^{-3 x}}{x^{13}}+\frac {96 e^{-2 x}}{x^{12}}-\frac {384 e^{-2 x}}{x^{11}}+\frac {384 e^{-2 x}}{x^{10}}+\frac {16 e^{-x}}{x^8}-\frac {32 e^{-x}}{x^7}-\frac {8388608 e^{-4 x}}{123743795175 x^6}+\frac {67108864 e^{-4 x}}{32564156625 x^5}+\frac {1}{x^4}-\frac {33554432 e^{-4 x}}{1206079875 x^4}+\frac {134217728 e^{-4 x}}{638512875 x^3}-\frac {536870912 e^{-4 x}}{638512875 x^2}-\frac {6561 e^{-3 x}}{7007000 x^2}+\frac {59049 e^{-3 x}}{700700 x}+\frac {59049 \text {Ei}(-3 x)}{25025}-\frac {16777216 \int \frac {e^{-4 x}}{x^7} \, dx}{41247931725}-\frac {6561 \int \frac {e^{-3 x}}{x^3} \, dx}{3503500}+\frac {1241513984 \int \frac {e^{-4 x}}{x^6} \, dx}{123743795175}+\frac {570807 \int \frac {e^{-3 x}}{x^2} \, dx}{7007000}-\frac {134217728 \int \frac {e^{-4 x}}{x^5} \, dx}{1302566265}+\frac {268435456 \int \frac {e^{-4 x}}{x^4} \, dx}{516891375}-\frac {536870912 \int \frac {e^{-4 x}}{x^3} \, dx}{638512875}-\frac {59049 \int \frac {e^{-3 x}}{x} \, dx}{28028}-\frac {2147483648 \int \frac {e^{-4 x}}{x^2} \, dx}{638512875}\\ &=\frac {256 e^{-4 x}}{x^{20}}-\frac {2048 e^{-4 x}}{x^{19}}+\frac {6144 e^{-4 x}}{x^{18}}-\frac {8192 e^{-4 x}}{x^{17}}+\frac {4096 e^{-4 x}}{x^{16}}+\frac {256 e^{-3 x}}{x^{16}}-\frac {1536 e^{-3 x}}{x^{15}}+\frac {3072 e^{-3 x}}{x^{14}}-\frac {2048 e^{-3 x}}{x^{13}}+\frac {96 e^{-2 x}}{x^{12}}-\frac {384 e^{-2 x}}{x^{11}}+\frac {384 e^{-2 x}}{x^{10}}+\frac {16 e^{-x}}{x^8}-\frac {32 e^{-x}}{x^7}+\frac {33554432 e^{-4 x}}{618718975875 x^5}+\frac {1}{x^4}-\frac {67108864 e^{-4 x}}{32564156625 x^4}+\frac {134217728 e^{-4 x}}{3618239625 x^3}-\frac {268435456 e^{-4 x}}{638512875 x^2}+\frac {2147483648 e^{-4 x}}{638512875 x}+\frac {19683 e^{-3 x}}{7007000 x}+\frac {177147 \text {Ei}(-3 x)}{700700}+\frac {33554432 \int \frac {e^{-4 x}}{x^6} \, dx}{123743795175}+\frac {19683 \int \frac {e^{-3 x}}{x^2} \, dx}{7007000}-\frac {4966055936 \int \frac {e^{-4 x}}{x^5} \, dx}{618718975875}+\frac {134217728 \int \frac {e^{-4 x}}{x^4} \, dx}{1302566265}-\frac {1712421 \int \frac {e^{-3 x}}{x} \, dx}{7007000}-\frac {1073741824 \int \frac {e^{-4 x}}{x^3} \, dx}{1550674125}+\frac {1073741824 \int \frac {e^{-4 x}}{x^2} \, dx}{638512875}+\frac {8589934592 \int \frac {e^{-4 x}}{x} \, dx}{638512875}\\ &=\frac {256 e^{-4 x}}{x^{20}}-\frac {2048 e^{-4 x}}{x^{19}}+\frac {6144 e^{-4 x}}{x^{18}}-\frac {8192 e^{-4 x}}{x^{17}}+\frac {4096 e^{-4 x}}{x^{16}}+\frac {256 e^{-3 x}}{x^{16}}-\frac {1536 e^{-3 x}}{x^{15}}+\frac {3072 e^{-3 x}}{x^{14}}-\frac {2048 e^{-3 x}}{x^{13}}+\frac {96 e^{-2 x}}{x^{12}}-\frac {384 e^{-2 x}}{x^{11}}+\frac {384 e^{-2 x}}{x^{10}}+\frac {16 e^{-x}}{x^8}-\frac {32 e^{-x}}{x^7}+\frac {1}{x^4}-\frac {33554432 e^{-4 x}}{618718975875 x^4}+\frac {268435456 e^{-4 x}}{97692469875 x^3}-\frac {268435456 e^{-4 x}}{3618239625 x^2}+\frac {1073741824 e^{-4 x}}{638512875 x}+\frac {8589934592 \text {Ei}(-4 x)}{638512875}+\frac {59049 \text {Ei}(-3 x)}{7007000}-\frac {134217728 \int \frac {e^{-4 x}}{x^5} \, dx}{618718975875}+\frac {4966055936 \int \frac {e^{-4 x}}{x^4} \, dx}{618718975875}-\frac {59049 \int \frac {e^{-3 x}}{x} \, dx}{7007000}-\frac {536870912 \int \frac {e^{-4 x}}{x^3} \, dx}{3907698795}+\frac {2147483648 \int \frac {e^{-4 x}}{x^2} \, dx}{1550674125}-\frac {4294967296 \int \frac {e^{-4 x}}{x} \, dx}{638512875}\\ &=\frac {256 e^{-4 x}}{x^{20}}-\frac {2048 e^{-4 x}}{x^{19}}+\frac {6144 e^{-4 x}}{x^{18}}-\frac {8192 e^{-4 x}}{x^{17}}+\frac {4096 e^{-4 x}}{x^{16}}+\frac {256 e^{-3 x}}{x^{16}}-\frac {1536 e^{-3 x}}{x^{15}}+\frac {3072 e^{-3 x}}{x^{14}}-\frac {2048 e^{-3 x}}{x^{13}}+\frac {96 e^{-2 x}}{x^{12}}-\frac {384 e^{-2 x}}{x^{11}}+\frac {384 e^{-2 x}}{x^{10}}+\frac {16 e^{-x}}{x^8}-\frac {32 e^{-x}}{x^7}+\frac {1}{x^4}+\frac {134217728 e^{-4 x}}{1856156927625 x^3}-\frac {536870912 e^{-4 x}}{97692469875 x^2}+\frac {1073741824 e^{-4 x}}{3618239625 x}+\frac {4294967296 \text {Ei}(-4 x)}{638512875}+\frac {134217728 \int \frac {e^{-4 x}}{x^4} \, dx}{618718975875}-\frac {19864223744 \int \frac {e^{-4 x}}{x^3} \, dx}{1856156927625}+\frac {1073741824 \int \frac {e^{-4 x}}{x^2} \, dx}{3907698795}-\frac {8589934592 \int \frac {e^{-4 x}}{x} \, dx}{1550674125}\\ &=\frac {256 e^{-4 x}}{x^{20}}-\frac {2048 e^{-4 x}}{x^{19}}+\frac {6144 e^{-4 x}}{x^{18}}-\frac {8192 e^{-4 x}}{x^{17}}+\frac {4096 e^{-4 x}}{x^{16}}+\frac {256 e^{-3 x}}{x^{16}}-\frac {1536 e^{-3 x}}{x^{15}}+\frac {3072 e^{-3 x}}{x^{14}}-\frac {2048 e^{-3 x}}{x^{13}}+\frac {96 e^{-2 x}}{x^{12}}-\frac {384 e^{-2 x}}{x^{11}}+\frac {384 e^{-2 x}}{x^{10}}+\frac {16 e^{-x}}{x^8}-\frac {32 e^{-x}}{x^7}+\frac {1}{x^4}-\frac {268435456 e^{-4 x}}{1856156927625 x^2}+\frac {2147483648 e^{-4 x}}{97692469875 x}+\frac {4294967296 \text {Ei}(-4 x)}{3618239625}-\frac {536870912 \int \frac {e^{-4 x}}{x^3} \, dx}{1856156927625}+\frac {39728447488 \int \frac {e^{-4 x}}{x^2} \, dx}{1856156927625}-\frac {4294967296 \int \frac {e^{-4 x}}{x} \, dx}{3907698795}\\ &=\frac {256 e^{-4 x}}{x^{20}}-\frac {2048 e^{-4 x}}{x^{19}}+\frac {6144 e^{-4 x}}{x^{18}}-\frac {8192 e^{-4 x}}{x^{17}}+\frac {4096 e^{-4 x}}{x^{16}}+\frac {256 e^{-3 x}}{x^{16}}-\frac {1536 e^{-3 x}}{x^{15}}+\frac {3072 e^{-3 x}}{x^{14}}-\frac {2048 e^{-3 x}}{x^{13}}+\frac {96 e^{-2 x}}{x^{12}}-\frac {384 e^{-2 x}}{x^{11}}+\frac {384 e^{-2 x}}{x^{10}}+\frac {16 e^{-x}}{x^8}-\frac {32 e^{-x}}{x^7}+\frac {1}{x^4}+\frac {1073741824 e^{-4 x}}{1856156927625 x}+\frac {8589934592 \text {Ei}(-4 x)}{97692469875}+\frac {1073741824 \int \frac {e^{-4 x}}{x^2} \, dx}{1856156927625}-\frac {158913789952 \int \frac {e^{-4 x}}{x} \, dx}{1856156927625}\\ &=\frac {256 e^{-4 x}}{x^{20}}-\frac {2048 e^{-4 x}}{x^{19}}+\frac {6144 e^{-4 x}}{x^{18}}-\frac {8192 e^{-4 x}}{x^{17}}+\frac {4096 e^{-4 x}}{x^{16}}+\frac {256 e^{-3 x}}{x^{16}}-\frac {1536 e^{-3 x}}{x^{15}}+\frac {3072 e^{-3 x}}{x^{14}}-\frac {2048 e^{-3 x}}{x^{13}}+\frac {96 e^{-2 x}}{x^{12}}-\frac {384 e^{-2 x}}{x^{11}}+\frac {384 e^{-2 x}}{x^{10}}+\frac {16 e^{-x}}{x^8}-\frac {32 e^{-x}}{x^7}+\frac {1}{x^4}+\frac {4294967296 \text {Ei}(-4 x)}{1856156927625}-\frac {4294967296 \int \frac {e^{-4 x}}{x} \, dx}{1856156927625}\\ &=\frac {256 e^{-4 x}}{x^{20}}-\frac {2048 e^{-4 x}}{x^{19}}+\frac {6144 e^{-4 x}}{x^{18}}-\frac {8192 e^{-4 x}}{x^{17}}+\frac {4096 e^{-4 x}}{x^{16}}+\frac {256 e^{-3 x}}{x^{16}}-\frac {1536 e^{-3 x}}{x^{15}}+\frac {3072 e^{-3 x}}{x^{14}}-\frac {2048 e^{-3 x}}{x^{13}}+\frac {96 e^{-2 x}}{x^{12}}-\frac {384 e^{-2 x}}{x^{11}}+\frac {384 e^{-2 x}}{x^{10}}+\frac {16 e^{-x}}{x^8}-\frac {32 e^{-x}}{x^7}+\frac {1}{x^4}\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 1.41, size = 23, normalized size = 0.92 \begin {gather*} \frac {e^{-4 x} \left (4-8 x+e^x x^4\right )^4}{x^{20}} \end {gather*}

Integrate[(-5120 + 37888*x - 102400*x^2 + 114688*x^3 - 32768*x^4 - 16384*x^5 - 4*E^(4*x)*x^16 + E^x*x^2*(-4096

*x^2 + 22272*x^3 - 38400*x^4 + 17408*x^5 + 6144*x^6) + E^(2*x)*x^4*(-1152*x^4 + 4032*x^5 - 3072*x^6 - 768*x^7)

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fricas [B] time = 1.01, size = 131, normalized size = 5.24 \begin {gather*} \frac {{\left (x^{8} e^{\left (4 \, x + 4 \, \log \left (x^{2}\right )\right )} + 4096 \, x^{4} - 8192 \, x^{3} + 6144 \, x^{2} - 16 \, {\left (2 \, x^{7} - x^{6}\right )} e^{\left (3 \, x + 3 \, \log \left (x^{2}\right )\right )} + 96 \, {\left (4 \, x^{6} - 4 \, x^{5} + x^{4}\right )} e^{\left (2 \, x + 2 \, \log \left (x^{2}\right )\right )} - 256 \, {\left (8 \, x^{5} - 12 \, x^{4} + 6 \, x^{3} - x^{2}\right )} e^{\left (x + \log \left (x^{2}\right )\right )} - 2048 \, x + 256\right )} e^{\left (-4 \, x - 4 \, \log \left (x^{2}\right )\right )}}{x^{12}} \end {gather*}

integrate((-4*x^8*exp(log(x^2)+x)^4+(32*x^8+208*x^7-128*x^6)*exp(log(x^2)+x)^3+(-768*x^7-3072*x^6+4032*x^5-115

2*x^4)*exp(log(x^2)+x)^2+(6144*x^6+17408*x^5-38400*x^4+22272*x^3-4096*x^2)*exp(log(x^2)+x)-16384*x^5-32768*x^4

+114688*x^3-102400*x^2+37888*x-5120)/x^13/exp(log(x^2)+x)^4,x, algorithm="fricas")

(x^8*e^(4*x + 4*log(x^2)) + 4096*x^4 - 8192*x^3 + 6144*x^2 - 16*(2*x^7 - x^6)*e^(3*x + 3*log(x^2)) + 96*(4*x^6

- 4*x^5 + x^4)*e^(2*x + 2*log(x^2)) - 256*(8*x^5 - 12*x^4 + 6*x^3 - x^2)*e^(x + log(x^2)) - 2048*x + 256)*e^(

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giac [B] time = 0.16, size = 129, normalized size = 5.16 \begin {gather*} \frac {x^{16} - 32 \, x^{13} e^{\left (-x\right )} + 16 \, x^{12} e^{\left (-x\right )} + 384 \, x^{10} e^{\left (-2 \, x\right )} - 384 \, x^{9} e^{\left (-2 \, x\right )} + 96 \, x^{8} e^{\left (-2 \, x\right )} - 2048 \, x^{7} e^{\left (-3 \, x\right )} + 3072 \, x^{6} e^{\left (-3 \, x\right )} - 1536 \, x^{5} e^{\left (-3 \, x\right )} + 256 \, x^{4} e^{\left (-3 \, x\right )} + 4096 \, x^{4} e^{\left (-4 \, x\right )} - 8192 \, x^{3} e^{\left (-4 \, x\right )} + 6144 \, x^{2} e^{\left (-4 \, x\right )} - 2048 \, x e^{\left (-4 \, x\right )} + 256 \, e^{\left (-4 \, x\right )}}{x^{20}} \end {gather*}

integrate((-4*x^8*exp(log(x^2)+x)^4+(32*x^8+208*x^7-128*x^6)*exp(log(x^2)+x)^3+(-768*x^7-3072*x^6+4032*x^5-115

2*x^4)*exp(log(x^2)+x)^2+(6144*x^6+17408*x^5-38400*x^4+22272*x^3-4096*x^2)*exp(log(x^2)+x)-16384*x^5-32768*x^4

(x^16 - 32*x^13*e^(-x) + 16*x^12*e^(-x) + 384*x^10*e^(-2*x) - 384*x^9*e^(-2*x) + 96*x^8*e^(-2*x) - 2048*x^7*e^

(-3*x) + 3072*x^6*e^(-3*x) - 1536*x^5*e^(-3*x) + 256*x^4*e^(-3*x) + 4096*x^4*e^(-4*x) - 8192*x^3*e^(-4*x) + 61

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method	result	size

risch	$\frac {1}{x^{4}}-\frac {16 \left (2 x -1\right ) {\mathrm e}^{-x}}{x^{8}}+\frac {96 \left (4 x^{2}-4 x +1\right ) {\mathrm e}^{-2 x}}{x^{12}}-\frac {256 \left (8 x^{3}-12 x^{2}+6 x -1\right ) {\mathrm e}^{-3 x}}{x^{16}}+\frac {256 \left (16 x^{4}-32 x^{3}+24 x^{2}-8 x +1\right ) {\mathrm e}^{-4 x}}{x^{20}}$	$91$
default	$\text {Expression too large to display}$	$36313$

int((-4*x^8*exp(ln(x^2)+x)^4+(32*x^8+208*x^7-128*x^6)*exp(ln(x^2)+x)^3+(-768*x^7-3072*x^6+4032*x^5-1152*x^4)*e

xp(ln(x^2)+x)^2+(6144*x^6+17408*x^5-38400*x^4+22272*x^3-4096*x^2)*exp(ln(x^2)+x)-16384*x^5-32768*x^4+114688*x^

1/x^4-16*(2*x-1)/x^8*exp(-x)+96*(4*x^2-4*x+1)/x^12*exp(-2*x)-256*(8*x^3-12*x^2+6*x-1)/x^16*exp(-3*x)+256*(16*x

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maxima [C] time = 0.46, size = 124, normalized size = 4.96 \begin {gather*} \frac {1}{x^{4}} - 32 \, \Gamma \left (-6, x\right ) - 208 \, \Gamma \left (-7, x\right ) + 128 \, \Gamma \left (-8, x\right ) + 393216 \, \Gamma \left (-9, 2 \, x\right ) + 3145728 \, \Gamma \left (-10, 2 \, x\right ) - 8257536 \, \Gamma \left (-11, 2 \, x\right ) - 3265173504 \, \Gamma \left (-12, 3 \, x\right ) + 4718592 \, \Gamma \left (-12, 2 \, x\right ) - 27753974784 \, \Gamma \left (-13, 3 \, x\right ) + 183666009600 \, \Gamma \left (-14, 3 \, x\right ) + 17592186044416 \, \Gamma \left (-15, 4 \, x\right ) - 319578856704 \, \Gamma \left (-15, 3 \, x\right ) + 140737488355328 \, \Gamma \left (-16, 4 \, x\right ) + 176319369216 \, \Gamma \left (-16, 3 \, x\right ) - 1970324836974592 \, \Gamma \left (-17, 4 \, x\right ) + 7036874417766400 \, \Gamma \left (-18, 4 \, x\right ) - 10414574138294272 \, \Gamma \left (-19, 4 \, x\right ) + 5629499534213120 \, \Gamma \left (-20, 4 \, x\right ) \end {gather*}

integrate((-4*x^8*exp(log(x^2)+x)^4+(32*x^8+208*x^7-128*x^6)*exp(log(x^2)+x)^3+(-768*x^7-3072*x^6+4032*x^5-115

2*x^4)*exp(log(x^2)+x)^2+(6144*x^6+17408*x^5-38400*x^4+22272*x^3-4096*x^2)*exp(log(x^2)+x)-16384*x^5-32768*x^4

+114688*x^3-102400*x^2+37888*x-5120)/x^13/exp(log(x^2)+x)^4,x, algorithm="maxima")

1/x^4 - 32*gamma(-6, x) - 208*gamma(-7, x) + 128*gamma(-8, x) + 393216*gamma(-9, 2*x) + 3145728*gamma(-10, 2*x

) - 8257536*gamma(-11, 2*x) - 3265173504*gamma(-12, 3*x) + 4718592*gamma(-12, 2*x) - 27753974784*gamma(-13, 3*

x) + 183666009600*gamma(-14, 3*x) + 17592186044416*gamma(-15, 4*x) - 319578856704*gamma(-15, 3*x) + 1407374883

55328*gamma(-16, 4*x) + 176319369216*gamma(-16, 3*x) - 1970324836974592*gamma(-17, 4*x) + 7036874417766400*gam

ma(-18, 4*x) - 10414574138294272*gamma(-19, 4*x) + 5629499534213120*gamma(-20, 4*x)

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mupad [B] time = 5.00, size = 88, normalized size = 3.52 \begin {gather*} \frac {1}{x^4}-\frac {{\mathrm {e}}^{-3\,x}\,\left (2048\,x^3-3072\,x^2+1536\,x-256\right )}{x^{16}}+\frac {{\mathrm {e}}^{-4\,x}\,\left (4096\,x^4-8192\,x^3+6144\,x^2-2048\,x+256\right )}{x^{20}}-\frac {{\mathrm {e}}^{-x}\,\left (32\,x-16\right )}{x^8}+\frac {{\mathrm {e}}^{-2\,x}\,\left (384\,x^2-384\,x+96\right )}{x^{12}} \end {gather*}

int(-(exp(- 4*x - 4*log(x^2))*(4*x^8*exp(4*x + 4*log(x^2)) - exp(x + log(x^2))*(22272*x^3 - 4096*x^2 - 38400*x

^4 + 17408*x^5 + 6144*x^6) - 37888*x - exp(3*x + 3*log(x^2))*(208*x^7 - 128*x^6 + 32*x^8) + 102400*x^2 - 11468

8*x^3 + 32768*x^4 + 16384*x^5 + exp(2*x + 2*log(x^2))*(1152*x^4 - 4032*x^5 + 3072*x^6 + 768*x^7) + 5120))/x^13

1/x^4 - (exp(-3*x)*(1536*x - 3072*x^2 + 2048*x^3 - 256))/x^16 + (exp(-4*x)*(6144*x^2 - 2048*x - 8192*x^3 + 409

6*x^4 + 256))/x^20 - (exp(-x)*(32*x - 16))/x^8 + (exp(-2*x)*(384*x^2 - 384*x + 96))/x^12

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sympy [B] time = 0.30, size = 97, normalized size = 3.88 \begin {gather*} \frac {1}{x^{4}} + \frac {\left (- 32 x^{49} + 16 x^{48}\right ) e^{- x} + \left (384 x^{46} - 384 x^{45} + 96 x^{44}\right ) e^{- 2 x} + \left (- 2048 x^{43} + 3072 x^{42} - 1536 x^{41} + 256 x^{40}\right ) e^{- 3 x} + \left (4096 x^{40} - 8192 x^{39} + 6144 x^{38} - 2048 x^{37} + 256 x^{36}\right ) e^{- 4 x}}{x^{56}} \end {gather*}

integrate((-4*x**8*exp(ln(x**2)+x)**4+(32*x**8+208*x**7-128*x**6)*exp(ln(x**2)+x)**3+(-768*x**7-3072*x**6+4032

*x**5-1152*x**4)*exp(ln(x**2)+x)**2+(6144*x**6+17408*x**5-38400*x**4+22272*x**3-4096*x**2)*exp(ln(x**2)+x)-163

84*x**5-32768*x**4+114688*x**3-102400*x**2+37888*x-5120)/x**13/exp(ln(x**2)+x)**4,x)

x**(-4) + ((-32*x**49 + 16*x**48)*exp(-x) + (384*x**46 - 384*x**45 + 96*x**44)*exp(-2*x) + (-2048*x**43 + 3072

*x**42 - 1536*x**41 + 256*x**40)*exp(-3*x) + (4096*x**40 - 8192*x**39 + 6144*x**38 - 2048*x**37 + 256*x**36)*e

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3.77.75 \(\int \frac {e^{-4 x} (-5120+37888 x-102400 x^2+114688 x^3-32768 x^4-16384 x^5-4 e^{4 x} x^{16}+e^x x^2 (-4096 x^2+22272 x^3-38400 x^4+17408 x^5+6144 x^6)+e^{2 x} x^4 (-1152 x^4+4032 x^5-3072 x^6-768 x^7)+e^{3 x} x^6 (-128 x^6+208 x^7+32 x^8))}{x^{21}} \, dx\)